I think Cat's treatment of Jon does make her look rather petty, because having been born and being where he was was never Jon's fault, so she's effectively bullying him in her position of (relative) authority over him.

That said, she is one of my favorite characters, I can understand why she felt the way she did even if I don't agree with how she expressed those feelings, and her behavior toward Jon actually made me like her more -- I just thought it made her deliciously well-rounded and human to give her such an obvious and glaring flaw. I do love her a lot.